K-G vs K

Unless you are in the bulldog chess club, you might not know what I am talking about

can a king and a guard mate a lone king?

Just pretend the rook is a guard. Although, I'm not sure this position is forcable.

King + Guard can force mate on all square boards up to size 14x14.

Beyond that, it is in general a draw (except for a very small fraction of all positions, where the bare King is already trapped in a corner).

It is interesting that the end-game King + Guard vs King + Guard is a forced win much more often than you would expect, on 8x8. When one player can cut off the Guard from its King, this Guard becomes basically useless; it has to be protected to pose any danger to the opposing King or Guard. You can then gang up on the defenceless King with your King + Guard. Having extra tempo moves with the Guard will not save him, as you can drive him back with checks.

Of what? There are 65 thousand different KGK positions (each in 8 orientations).

Better just practice it agains an engine. E.g. by setting up a position for Courier Chess. (Which is on a 12x8 board, but that should not matter much for the technique; it just takes a bit longer.) Let the computer play K+G and try to defend it.

Really?

I haven't had time to study this, but this answer is a bit surprising. Why just square boards, and why not larger than 14 x 14?

I know RK is a win against K. So I'm assuming the strategy of GK vs K may have similarities in strategy (but more complicated and take longer in general due to value of guard lower than rook).

I'm also surprised this result is sensitive to board size, and that it might not be the same for square boards vs. rectangular boards.

On the other hand, usually HGMuller knows what he's talking about.😐

On a big enough board rook and king is drawn because of the fifty move rulle, it's probably the same with king and guard (just guessing but it would make sense)

For those who don't know, HGMuller has programmed chess engines which can play variant chess pieces. But as far as I know, nobody has written an engine that can play games as fast as new ones are invented. That would obviously be a futile effort. Nevertheless, HGMuller's software is really useful to answer questions about guards against kings (and stuff like that)

Good point. So the answer could depend on the initial position of pieces. I hope nobody asks to see the answer to them all.

This is somewhat unethical. You should be asking these questions AFTER the game if you fail to convert.

It's surprising you need to ask, your rating is 2000 (better than me) yet you can't use your intuition to figure out how to force a win with K+G vs K? I could win such an endgame easily, assuming no 50-rule move (not even sure if we play that in bulldog). With the 50-move rule, I'd still be confident of converting on a standard bulldog board. And that's before I read Muller's comments.

Agree. Asking for help and using engines to analyze an ongoing game is cheating. Rules are (here).

If anyone has evidence of anyone who has done this to get an advantage please bring it to the (bulldog chess players club). Those games will probably be erased from the score keeping system, and those players will not be allowed to compete in prize games.😐

It's possible that because English isn't his first language, that there's something lost in translation and he's not actually asking regarding an ongoing endgame.

If he does have a K+G vs K endgame here, I suggest calling it a draw and it should be considered a warning.

Oh wait, ry is American. For some reason I thought he was Hungarian, but that's Muller.

edit - ha no Muller is Dutch. Why have I got Hungary in my head?

This is not primarily a 50-move issue; even in complete absence of such a rule short-range pieces tend to lose their mating potential on large boards. On large board it doesn't make much sense to keep the allowed number of moves as low as 50 anyway; this would spoil many forced wins. In orthodox Chess this number was picked because if the win can be forced, it can almost always be forced within 35 moves, so that you did not make progress for 50 moves is a very strong indication the game is indeed a dead draw.

The point is that neither a King or a Guard alone can confine a King. So to prevent it running to safety you always needs the pair. But the pair moves twice as slow as the bare King, because you are not allowed to move both in the same turn. So the bare King continuously outruns the pair. That doesn't mean it is always hopeless: the pair covers an area of a certain size, and the bare King would have to encircle that to get to safety. The total width of the covered area can be 6 squares. So the bare King has to gain 6 steps on the pair, and each 2 turns it only gains 1 step. So it takes ~12 turns for the bare King to overtake its persuitors, in which he can cover a distance of 12. If he runs into an edge before that, he is confined (and then gets trapped against that edge, so that trying to outrun the K+G is not the best defensive strategy). But if the edge is further away, he then moves towards freedom once he has left the pair behind.

So the defensive strategy against a set of short-range pieces is to use the board as a race track, running circles along the edge to get clear of yoiur pursuitors, and move towards the center when you get ahead of them (or have dispersed them, so those near you do no longer form an effective unit).

A Rook can confine a King virtually without moving (you have to switch now end then to the opposite edge when you get attacked), so there is no way to outrun it. The strong side can use nearly all his moves on his King, to approach it and confine the bare King in the other dimension.

A similar situation occurs in the 4-men ending King + Bishop + weak piece vs. King. Unlike the Rook, the Bishop cannot confine a King. But King + Bishop together can. The Bishop hardly needs any moves for this confinement; it covers a diagonal, and the attacking King has to cover the square where the bare King could sneak through. So you basically have a stationary Bishop and King vs King, the latter of course having equal speed. So the attacking King can just keep up with the bare King. The point is that the bare King cannot reverse its direction without losing a tempo; to maintain the threat of crossing the diagonal, it has to keep moving in the same direction. Where it sooner or later will encounter an edge, so that it has to reverse. At that point the strong player has a 'free move' to approach his weak piece. So sooner or later that weak piece gets to the action, and with the aid of it wou will be able to drive the King away from the diagonal, and then tighten the Bishop noose. So this end-game is a forced win on any size board, but it can take excessively long.

As to board size: that I mention square boards does not mean it cannot be done on rectangular boards. Just that I never tried it. Obviously when you can do it on an NxN board, you can also do it on NxM or MxN with M < N. But if you cannot force mate on NxN, you often can force mate on MxL, with M<N and L>N. E.g. on 15x15 KGK is almost always draw, but on 8x100 it is a very easy (but very lengthy) win. The point is that you can cut confine the bare King along the narrow dimension, and then slowly but surely push him to the edge along the length of the board.

This is an example of an actual mating line; you can see that the mate is very straightforward, based on a pincher movement pushing the bare King to a corner by zugzwang. I represented the Guard by a Queen so that all Guard moves are considered at least pseudo-legal. I still cannot publish it here as a game, because the viewer software here imagines moves on the Guard that would deliver check, so it does consider moves of the bare King not resolving such phanto checks, or wandering into them illegal (and refuses them).

The game was recorded by Fairy-Max self-play from the given position, having the bare King play at 40 moves/hour, and the strong side at 40 moves/6 min.

[Event "Computer Chess Game"]
[Site "MAKRO-PC"]
[Date "2018.05.07"]
[Round "-"]
[White "Fairy-Max 5.0b6"]
[Black "Fairy-Max 5.0b6"]
[Result "1-0"]
[TimeControl "40/3600"]
[Variant "guard"]
[VariantMen "Q:K"]
[FEN "8/8/8/8/8/1k6/8/1K5Q w - - 0 1"]
[SetUp "1"]

{--------------
. . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
. k . . . . . .
. . . . . . . .
. K . . . . . Q
white to play
--------------}
1. Qg2 Kc3 2. Qf3 Kd3 3. Qf4 Kc4 4. Kc2 Kd4 5. Qf5 Kc5 6. Kc3 Kd5 7. Kd3
Kc6 8. Qe6 Kc5 9. Kc3 Kc6 10. Kc4 Kb7 11. Qd6 Kc8 12. Kd5 Kb7 13. Kc5 Ka6
14. Qc6 Ka7 15. Qc7 Ka8 16. Kc6 Ka7 17. Qb7#
{Xboard adjudication: Checkmate} 1-0

How absolutely fascinating. I assumed it was a 50-move issue, but that makes perfect sense.

It's sunny, I've been up sice 5.30am, and I'm baked.

Thanks for the thorough explanation @HGMuller, I found it really interesting to read.